C.1: Develop mathematical thinking and communication skillsCourses designed for mathematical sciences majors should ensure
that students

Progress from a procedural/computational understanding of mathematics
to a broad understanding encompassing logical reasoning, generalization, abstraction,
and formal proof;

...

While I have been devoting one column to each of the recommendations in the
CUPM Curriculum Guide, this recommendation is so rich and so important that
I have decided to write a column on each of its three parts. This month, I am
focusing on the transition to proof. In May I will consider the demand that
all majors learn how to analyze data. For June, I will tackle the huge subject
of oral and written communication of mathematics.

When thinking about the first-year/sophomore mathematics courses that will
be taken by potential majors in the mathematical sciences, the development of
mathematical thinking is what it is all about. Actually, that is not unique
to those who will major in our subject. When we held the Curriculum Foundations
Project workshops, that was also the message that we heard from chemists, physicists,
engineers, in short everyone with a substantial mathematical requirement of
their majors. This point was particularly well articulated by the participants
from graduate programs in Mathematics at their workshop, held at MSRI in February,
2001:

“The most important task of the first two years is to move students
from a procedural/computational understanding of mathematics to a broad understanding
encompassing logical reasoning, generalization, abstraction, and formal proof.
This should be accomplished as soon as possible in a student’s undergraduate
career.”[1, p. 109]

It has been my experience that this emphasis is not only beneficial to students,
it is what energizes and excites them about mathematics, luring them into taking
additional courses. Again, the Mathematics workshop at MSRI made this point:
“If you want to recruit, teach good courses. We lose a lot of students
who come in thinking that they want to be math majors by failing to make real
to them the intellectual vitality of mathematics.”[1,
p. 112]

Every course must have as one of its conscious goals the development of mathematical
thinking and communication skills. This must be seen as a continuing process
that begins with the first college course in mathematics and continues right
through the last. The level and extent to which this is done must be appropriate
to the students in the course. Introductory real analysis in place of calculus
is appropriate for at most a very tiny slice of the student population, and
then only for those who already understand the basic techniques and concepts
of calculus. But even within a general calculus class, there are many opportunities
for investigating mathematical arguments and for dissecting the meaning and
purpose of definitions.

Students cannot learn how to construct mathematical proofs in a single “bridge”
course, just as they cannot learn the effective writing of mathematics in a
single course that has been designated with a “W.” However, bridge
courses can have a very important role to play, often offering a less structured
syllabus so that there is flexibility in choosing topics that engage student
interest and illustrate the role of proof within mathematics. Such courses can
develop proofs within a context that is familiar so that students can focus
on the logic of the argument rather than having to deal with significant new
terminology and concepts. Although they cannot carry the full weight of the
transition “from a procedural/computational understanding of mathematics
to a broad understanding” by themselves, bridge courses can usefully serve
as a piece of the ongoing process of developing mathematical thinking.

We need to worry about what it means to “teach proof.” Students
who cannot dissect someone else’s proof or discover a logically sound
proof of a result they have not seen before have not “learned proof.”
There is a lot of research on the constituents of this skill and what impedes
its development. The Illustrative
Resources includes links to and descriptions of work by Annie and John Selden,
Eric Knuth, Robert Moore, Susanna Epp and others on common student misconceptions
and difficulties with proof. The Illustrative Resources also point to many successful
programs at various colleges and universities, from Inquiry Guided Learning
and the Moore Method to problem-solving seminars and the workshops developed
at Rutgers.The Illustrative Resources also includes descriptions of and links
to specific recommendations by Susanna Epp, Ed Sandefur, and Moira McDermott
on how to teach proof.

For too many students, proofs are a game played with arbitrary rules for an
unclear purpose. They are simply one more bit of mathematics that must be memorized,
something totally divorced from the doing of mathematics. Our challenge is to
help students see the analysis and construction of proofs as a piece in the
process of mathematical argument and reasoning, as a means of tying new mathematical
knowledge to what was known before. Proofs test, clarify, and solidify our mathematical
knowledge. They are a piece of what makes mathematics exciting, surprising,
and satisfying.

We would appreciate more examples that document experiences with the use of
technology as well as examples of interdisciplinary cooperation.

David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester
College in St. Paul, Minnesota, he was one of the writers for the Curriculum
Guide, and he currently serves as Chair of the CUPM. He wrote this column
with help from his colleagues in CUPM, but it does not reflect an official
position of the committee. You can reach him at bressoud@macalester.edu.